(This section is sure to be ugly in the ASCII version of this document. For a pretty .dvi or .ps presentation of the underlying .tex file, go through my web site.)
Given an alphabet
(of variables 
, constants 
, n-ary
relation symbols 
, functors 
,
quantifiers 
, and the familiar truth-functional
connectives (
) one uses
standard formation rules (e.g., if 
and 
are well-formed
formulas, then 
is a wff as well) to build ``atomic"
formulas, and then more complicated ``molecular" formulas. Sets of these
formulas (say 
), given certain rules of inference (e.g., modus
ponens: from 
and 
infer to 
), can
lead to individual formulas (say 
); such a situation is expressed
by meta-expressions like 
. First-order logic, like
all logical systems, includes a semantic side which systematically
provides meaning for formulas involved. In first-order logic, formulas
are said to be true (or false) on an interpretation, often written
as 
. (This is often
read, ``
satisfies, or models,
.") For example, the formula 
might mean,
on the standard interpretation
for arithmetic, that for every natural number
n, there is a natural number m such that m > n. In this case,
the domain of 
is N, the natural numbers, and G is
the binary relation 
, i.e., > is
a set of ordered pairs 
where 
and i is
greater than j.
In order to concretize things a bit,
consider an expert system designed to play the role of a
guidance counselor in advising a high school student about which colleges
to apply to (I have such a system under development at Rensselaer). Suppose
that we want a rule in such a system which says ``If a student has
low SATs, and a low GPA, then none of the top twenty-five national universities
ought to be applied to by this student." Assume that we have
the following interpreted predicates: Sx iff x is a student,
for x has low SATs, 
for x
has a low GPA, Tx for x is a top twenty-five national university,
Axy for x ought to apply to y. Then the rule in question, in first-order
logic, becomes
Let's suppose, in addition, that Steve is a student denoted by the constant s in the system, and that he, alas, has low SATs and a low GPA. Assume also that v is a constant denoting Vanderbilt University (which happens to be a top twenty-five national university according the U.S. News' annual rankings). These facts are represented in the system by
and
Let's label these three facts, in the order in which they were presented, (1), (2), and (3). Our expert system, based as it is on first-order logic, can verify
that is, it can deduce that Steve ought not to apply to Vanderbilt.